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Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise

Binjie Li, Qin Zhou

Abstract

This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial approximation. A pathwise uniform convergence rate is derived, encompassing general spatial \( L^q \)-norms, by using discrete versions of deterministic and stochastic maximal \( L^p \)-regularity estimates. Additionally, the theoretical convergence rate is validated through numerical experiments. The primary contribution of this work is the introduction of a technique to establish the pathwise uniform convergence of finite element-based full discretizations for nonlinear stochastic parabolic equations within the framework of general spatial \( L^q \)-norms.

Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise

Abstract

This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial approximation. A pathwise uniform convergence rate is derived, encompassing general spatial -norms, by using discrete versions of deterministic and stochastic maximal -regularity estimates. Additionally, the theoretical convergence rate is validated through numerical experiments. The primary contribution of this work is the introduction of a technique to establish the pathwise uniform convergence of finite element-based full discretizations for nonlinear stochastic parabolic equations within the framework of general spatial -norms.
Paper Structure (13 sections, 9 theorems, 119 equations, 2 figures)

This paper contains 13 sections, 9 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Setup
  3. Full Discretization
  4. Proofs
  5. Preliminary lemmas
  6. A stability estimate for a discrete stochastic convolution
  7. Proof of
  8. Proof of the error estimate
  9. Proof of the stability property
  10. Proof of the error estimate
  11. Proof of the error estimate
  12. Numerical results
  13. Conclusions

Key Result

Proposition 2.1

Suppose the initial value $v$ belongs to $W_0^{1,\infty}(\mathcal{O}) \cap W^{2,\infty}(\mathcal{O})$. Then, the model problem eq:model admits a unique mild solution. Moreover, for any $p \in (2, \infty)$, $q \in [2, \infty)$, and $\epsilon > 0$, the solution $y$ satisfies

Figures (2)

  • Figure 1: $\tau=1.0\times 10^{-6}$, $T=0.01$,
  • Figure 2: $T=0.1$, $h=\tau^{1/2}$

Theorems & Definitions (14)

  • Proposition 2.1
  • Theorem 3.1
  • Remark 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • Proposition 4.1
  • ...and 4 more